{
  "abstract": "Convergence in parameter estimation classically requires ``persistency of excitation,\" which is a non-degeneracy condition on a trajectory-dependent signal. This paper develops, to the best of our knowledge, the first such excitation theory for a nonlinear partial differential equation. In the context of identifying the unknown viscosity from spectral observations in the two-dimensional Navier--Stokes equations for incompressible fluids, our excitation condition is computable, verifiable a priori, and sharp with respect to scaling. Our approach employs a data assimilation methodology to account for an unknown initial state and select candidate viscosities by minimizing the loss between the low-mode observations of the fluid velocity and low-mode projection of a nudging-based filter that assimilates these observations. The main novelty of our framework is to distinguish a new functional, $\\mathsf{W}$, representing the work done by the filter's associated sensitivity variable on the enstrophy, around which our entire analysis is centered. We show that $\\mathsf{W}$ is explicitly comparable to the observability Gramian of the associated Gauss--Newton iteration, and subsequently establish the following dichotomy at every critical point of the observational loss: either $\\mathsf{W}$ exceeds a certain threshold, in which case the candidate viscosity obeys an explicit error estimate that depends inversely on $\\mathsf{W}$ and the observational density $N$, but directly on the error between initial conditions, or else $\\mathsf{W}$ is below the threshold and the observations are quantitatively insensitive to parameter updates in a way that is detectable to the user. Notably, our approach is energy-based, and therefore expected to be adaptable to many other nonlinear dissipative systems.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.31230",
  "author_total": 3,
  "authors": [
    {
      "name": "Vincent R. Martinez"
    },
    {
      "name": "Sarah Strikwerda"
    },
    {
      "name": "Xiang Wan"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/03k3khakvn593.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.31230",
  "fulltext_url": null,
  "id": "03k3khakvn593",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.31230",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AP"
    ],
    [
      "arxiv",
      "math.OC"
    ],
    [
      "openalex",
      "2604|Navier-Stokes equation solutions"
    ]
  ],
  "openalex": "W7214576561",
  "pdf_url": "https://arxiv.org/pdf/2609.31230",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-25",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/03k3khakvn593.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:57Z",
      "native_id": "2609.31230",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.31230"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "Excitation and Identifiability in the 2D Navier-Stokes equations",
  "title_text": "Excitation and Identifiability in the 2D Navier-Stokes equations",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:22Z",
  "url": "https://research.draconitic.com/paper/03k3khakvn593",
  "venue": "arXiv",
  "versions": []
}